UDC 517.925

MATHEMATICS

D. M. GROΒMAN

TOPOLOGICAL EQUIVALENCE OF DYNAMICAL SYSTEMS

(Presented by Academician I. G. Petrovskii, 21 X 1966)

In this note theorems are formulated which are a generalization and development of results of the author \((^{1,2})\) and of Hartman \((^{3,4})\), concerning the topological equivalence of the systems

\[ \dot{x}=Ax+F(x), \tag{1} \]

\[ \dot{y}=Ay, \tag{2} \]

where \(A\) is a constant \(n\times n\) matrix; \(x, y, F\) are vectors of the \(n\)-dimensional space \(L^n\).

\(1^\circ\). We introduce the following definitions. A homeomorphism \(\Phi\) mapping some set \(G\subset L^n\) onto some set \(M\subset L^n\) in such a way that solutions of system (1) from \(G\) are transformed into solutions of system (2) from \(M\) and conversely will be called an \(H\)-homeomorphism of systems (1) and (2) on the sets \(G\) and \(M\), and the systems (1) and (2) will be called homeomorphic on \(G\) and \(M\).

If the \(H\)-homeomorphism \(\Phi\) has the form \(\Phi(x)=x+\varphi(x)\), where the vector \(\varphi(x)\) is bounded in norm \((|\varphi(x)|<\mathrm{const})\) for \(x\in G\), then we shall say that \(\Phi\) has a bounded displacement.

If the displacement \(\varphi\) of the \(H\)-homeomorphism \(\Phi\) for all \(x\in G\) satisfies the inequality \(|\varphi(x)|\le ac\), where \(a>0\) is determined by the matrix \(A\), and \(c=\sup_{x\in G}|F(x)|<+\infty\), then \(\Phi\) will be called an \(A\)-homeomorphism.

\(2^\circ\). Suppose the following:

a) the matrix \(A\) has no eigenvalues with zero real part;

b) the vector \(F(x)\) is defined and bounded in \(L^n\): \(|F(x)|\le c\) for any \(x\).

Theorem 1. Suppose that conditions a), b) are fulfilled and, in addition, the requirements are satisfied:

c) \(F(x)\) is continuous in \(L^n\) and ensures uniqueness of the solution of the Cauchy problem for system (1);

d) the difference of any two solutions of system (1) is unbounded on the \(t\)-axis.

Then

1) there exists an \(A\)-homeomorphism \(\Phi\) of systems (1) and (2), mapping \(L^n\) onto itself;

2) the homeomorphism \(\Phi\) is the unique \(A\)-homeomorphism of systems (1) and (2) in \(L^n\), and even the unique \(H\)-homeomorphism of systems (1) and (2) in \(L^n\) with bounded displacement;

3) condition d) is necessary for the existence in \(L^n\) of an \(H\)-homeomorphism of systems (1) and (2) with bounded displacement;

4) there exists an infinite set of \(H\)-homeomorphisms of systems (1) and (2).

Theorem 2. If requirements a) and b) are fulfilled, and the vector \(F(x)\) satisfies in \(L^n\) a Lipschitz condition with a sufficiently small constant, then assertions 1), 2), and 4) of Theorem 1 are true.

Theorem 3. If condition a) is satisfied, and the vector \(F(x)\) satisfies the Lipschitz condition in some bounded domain \(G \subset L^n\) with a sufficiently small constant, then there exists an \(A\)-homeomorphism of the systems (1) and (2) mapping the domain \(G\) onto some domain \(M\) of the space \(L^n\).

Theorem 4. If condition a) is fulfilled, and the vector \(F(x)\) satisfies the Lipschitz condition in \(L^n\) with a sufficiently small constant, then the systems (1) and (2) are homeomorphic in \(L^n\).

3°. Here we shall formulate a theorem on the homeomorphism of abstract dynamical systems. In doing so we shall use the terminology and notation adopted in the book [5].

Let a dynamical system \(\{R, f(p,t)\}\) be given, where \(R\) is a metric space, \(p\) is a point of \(R\), \(t \in (-\infty,+\infty)\), and \(f(p,t)\) is a one-parameter group of mappings (motions) of \(R\) onto itself.

We shall say that the point \(f(p,t')\) is an entry point, and the instant \(t=t'\) the entry time of the motion \(f(p,t)\) into a certain closed set \(M\), if \(f(p,t') \in M\), but for \(t < t'\), \(f(p,t) \notin M\). Analogously we define the exit point and the exit time \(f(p,t)\) from \(M\).

Since motions whose trajectories coincide cannot have different entry points into \(M\) (exit points from \(M\)), it makes sense to speak of entry points into \(M\) (exit points from \(M\)) of trajectories.

A closed set \(M\) will be called separating for the dynamical system \(\{R,f\}\) if:

1) every trajectory of the system \(\{R,f\}\) intersects \(M\);

2) if not the whole trajectory lies in \(M\), then either the trajectory has an entry point \(p^*\) into \(M\) and the positive semitrajectory \(f(p^*, I^+) \subset M\); or the trajectory has an exit point \(q^*\) from \(M\) and the negative semitrajectory \(f(q^*, I^-) \subset M\); or the trajectory has both an entry point into \(M\) and an exit point from \(M\), and the arc of the trajectory enclosed between these points belongs to \(M\);

3) for the motion \(f(p,t)\), the entry time into \(M\) (exit time from \(M\)), if it exists, depends continuously on \(p\).

Obviously, the concept of an \(H\)-homeomorphism is also applicable to abstract dynamical systems.

Theorem 5. Suppose that each of the dynamical systems \(\{R,f_i\}\), \(i=1,2\), has a separating set \(M_i\), and suppose that there exists an \(H\)-homeomorphism \(\Phi^*\), \(\Phi^*(M_1)=M_2\), of the systems \(\{R,f_1\}\) and \(\{R,f_2\}\).

Then one can construct an \(H\)-homeomorphism \(\Phi\) of the systems \(\{R,f_1\}\) and \(\{R,f_2\}\), mapping \(R\) onto itself and coinciding with \(\Phi^*\) on \(M_1\).

Obviously, the meaning of Theorem 5 is that it asserts the possibility of extending an \(H\)-homeomorphism from \(M_1\) to \(R\).

Theorem 4 is obtained from Theorem 2 with the aid of Theorem 5.

Institute of Electronic
Control Machines

Received
19 X 1966

REFERENCES

1. D. M. Grobman, DAN, 128, No. 5, 880 (1959).
2. D. M. Grobman, Matem. sborn., 56 (98), 1, 77 (1962).
3. P. Hartman, Proc. Am. Math. Soc., 11, No. 4, 610 (1960).
4. P. Hartman, Proc. Am. Math. Soc., 14, No. 4, 568 (1963).
5. V. V. Nemytskii, V. V. Stepanov, Qualitative Theory of Differential Equations, Moscow–Leningrad, 1949.